Re: is sci.math slow? correction to my post to the AP thread

In article <7935845b-0ba2-4eaa-aa32-3f9dab0c1265@xxxxxxxxxxxxxxxxxxxxxxxxxxx> Ken Quirici <ken.quirici@xxxxxxxxxx> writes:
....
I misread the mathworld def. of hyperbolic geometry - it does not say
there are more than one line thru a point parallel to a line not
containing the point. It says MANY such lines.

That makes sense. There will not be two, or three, or some other number
of parallel lines, but there will be infinitely many.

So I wonder if that means there is NO hyperbolic geometry that has
a point and a line such that there are exactly TWO lines thru
the point parallel to the line?

Indeed. If there are two lines through a point parallel to a given line,
there are infinitely many lines parallel through the point parallel to
the given line. Consider a simple model of a hyperbolic geometry: the
inside of a circle. Straight lines are the same as the Euclidean
straight lines. Now given a straight line and a point not on that line,
we can draw from that point lines connecting the point to the two
points were the given straight line crosses the circle, and there are
obviously two of them. Clearly, within the hyperbolic geometry they do
not cross the given line (the crossing points are on the circle, and that
is outside the geometry). But also all lines through that point and
between those two lines do not cross the given line.
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